Essay review of “The Laws of Belief. Ranking Theory and its Philosophical Applications” by Wolfgang Spohn. Oxford: Oxford University Press, 2012. 625 pages. £75. Hardback. ISBN 9780199697502. PENULTIMATE VERSION: PLEASE CITE THE OF SCIENCE PAPER!

Wolfgang Spohn’s book The Laws of Beliefs. Ranking Theory and its Philosophical Applications is a detailed presentation of ranking theory and its many and varied applications in philosophy. It is work in formal or mathematical philosophy, philosophy that uses formal or mathematical tools to tackle philosophical problems. The book uses one simple and elegant, yet powerful formal tool, namely the theory of ranking functions, and applies it fruitfully to a great variety of philosophical problems. These start with problems in such as the representation of belief and its objects as well as justification and the a priori. They go to problems in such as causation and objectification and problems in the such as laws and their confirmation as well as ceteris paribus conditions and dispositions. And they include, at least indirectly, problems in the such as conditionals and problems in the such as perception and consciousness. The result is a unified picture of the world and our relation to it as epistemic agents: a philosophy. It is arrived at in a constructive attempt to reduce all “natural” or alethic modalities to non-modal facts on the one hand and doxastic modalities as represented by ranking functions (and subjective probability measures) on the other hand. And it does so by displaying the virtues of using formal tools to tackle philosophical problems. The result is a formal philosophy whose radical nature, however, becomes visible only by working through the details, not something one can do over an extended weekend.

What I consider to be one of its most problematic features is the downside of the comprehensiveness of Spohn’s monumental piece: its length is scary, and many parts of the book are not an easy read, especially since it contains several new theorems that Spohn proves in the main body of text. Minimally, Spohn should have put the proofs in an appendix. Ideally he would have outsourced the proofs to technical papers in order to make the book more accessible in both length and style. While, as readers, we should not cherish the illusion that philosophy is easy, as authors we should try our best to not add to the difficulty. The plot of the book can be split up into three acts, and I think Spohn should have written three books rather than one, or at least he should have cut it up into three digestible volumes. That is, at least, how I want to perform the play: the first act corresponds to chapters 1-6 and 8-9; the second act roughly corresponds to chapters 10-13 and 16-17; the third act roughly corresponds to chapters 7 and 14-15 as well as some elements of chapter 12.

Ranking theory

Spohn arrives at the central notion of a ranking function, originally called “ordinal conditional function” in Spohn (1988), as the natural result of combining the frustrations of Bayesians to capture the notion of (categorical or outright or yes-or-no) belief (or belief simpliciter) in probabilistic terms and the frustrations of belief revision theorists in the AGM tradition (Alchourrón & Gärdenfors & Makinson 1985) to cope with iterated belief changes. The first thesis Spohn submits is that subjective probability theory in the Bayesian tradition does not have a notion of belief that is capable of being true (p. 44), although it has a dynamics that is complete (pp. 37ff). The latter is the case if regularity is imposed as an additional constraint on probability measures and Jeffrey conditionalization is adopted as update rule, or if classical probabilities are replaced by Popper-Rényi measures (Popper 1955, Rényi 1955) and these are improved upon as indicated in Spohn (1986) (see also pp. 202ff). The second thesis Spohn submits is that AGM belief revision theory does not have a dynamics that is even moderately complete (pp. 65ff), although it has a notion of belief that is capable of being true. In a nutshell the reason is that AGM belief revision theory does not have a notion of conditional belief and thus lacks the corresponding notions of (conditional) relevance and (conditional) independence. Bayesians and belief revision theorists alike do well to study this discussion.

In order to have a notion of belief that is capable of being true and as well as a complete dynamics for this notion that can handle indefinite iterations of belief changes one has to represent the agent’s doxastic state by a ranking function. Only in this way the agent’s conditional beliefs can be adequately captured. Here is a slightly revised version of Spohn’s definition 5.5 with notation adapted (p. 70): let A be a complete algebra of propositions over the set of possibilities or possible worlds W. Then  is a ranking function on A if and only if  is a function from W into the extended natural numbers N+ = N{} such that at least one possibility w from W is assigned rank 0.  is extended to propositions A from A by defining () =  and (A) = min {(w): w  A} for each non-empty A from A. The rank of a proposition A, (A), is interpreted as the agent’s grade of disbelief for A. The agent believes a proposition A just in case she assigns a positive degree disbelief to the negation of A, W\A: (W\A) > 0. The defining clause that at least one possibility be assigned rank 0 requires the agent to not disbelieve every possibility, and thus to not disbelieve every proposition. In other words, the static law of belief encapsulated in the definition of a ranking function is consistency.

The dynamic law of belief is stated in terms of conditional ranks, which definition 5.15 defines as differences of unconditional ranks (p. 78): (w|A) = (w) – (A) if w  A, and  otherwise, provided (A) < . The agent believes A conditional on C just in case she assigns a positive degree of disbelief to the negation of A, W\A, conditional on C: (W\A|C) > 0. Definition 5.24 then states what is often called Spohn conditionalization, the rank-theoretic analogue to Jeffrey conditionalization in probability theory (p. 83, in slightly revised form and with notation adapted): let  be a ranking function on A and let A be a proposition from A such that (A), (W\A) < , and let n be a number from N+. Then the An-conditionalization

An of  is defined by An(w) = (w|A) if w  A, and An(w) = (w|W\A) + n if w  W\A. Spohn conditionalization preserves and generalizes AGM belief revision theory (pp. 88ff) and provides a complete dynamics. It can be shown to adequately handle indefinite iterations of belief changes (pp. 171ff based on Hild & Spohn 2008), which at the same time answers the question where the numbers come from by providing a theory of measurement for ranks in terms of what belief revision theorists call “contractions” (belief changes where the old beliefs are contracted to make them consistent with some new belief). I personally prefer a different argument for the thesis that an agent should update her beliefs by Spohn conditionalization: the disadvantage is that I have to presuppose a way of measuring ranks; the advantage is that we can formulate the dynamic law of belief as diachronic consistency so that the “laws of belief” from the title of the book, as formulated in ranking theory, require nothing but consistency in a synchronic as well as a diachronic sense (Huber 2007).

The philosophical applications of ranking theory

Space does not permit to discuss more than one philosophical application of ranking theory. This application, though, is so important that it would have deserved a book on its own. It is Causation

What I consider to be Spohn’s most radical philosophical thesis could have played a more prominent role in his book. In its general form the view is only hinted at a few places (pp. 12ff, 271ff, 442ff, 469ff). In its general content the view is not new, as it has been held by David Hume on at least one interpretation. However, in the particular form it is spelt out for the particular problem at hand it is genuinely revolutionary, yet, I fear, not well-received.

Spohn’s view is that all there is are non-modal facts and doxastic modalities, a view he shares with Lewis. Lewis’ program of Humean supervenience was to show that all alethic modalities such as counterfactuality, causation, lawhood, chance etc. supervene on non- modal “local matters of particular fact” (Lewis 1986). Spohn thinks this does not work (pp. 272; Spohn 2010). Rather what he calls “natural” modalities – causation, lawhood, dispositions, chances – are the results of projecting our doxastic modalities – belief and degrees of belief – onto the non-modal facts: we are beefing up a world that is much more meager than we believe it to be. Spohn represents the doxastic modalities by ranking functions and subjective probability measures. The question is what it means that we project our thus represented doxastic modalities onto the non-modal facts. Spohn spells out the projectivist metaphor for his theory of causation (ch. 14; Spohn 2005) in constructive detail and with formal precision. The main thought is the following one (ch. 15; Spohn 1993): causation is defined relative to a subjectively interpreted ranking function, but some features of this subjective ranking function can be brought into one-to-one correspondence with certain non-modal propositions that are objectively true or false. For instance, the unconditional beliefs of a subjective ranking function can be brought into one-to-one correspondence with certain non-modal propositions that are objectively true or false, viz. the contents of those unconditional beliefs. Therefore unconditional belief, as defined in rank-theoretic terms, can be objectified (pp. 445ff). Each feature of a ranking function that can be uniquely associated with a non-modal proposition can itself be said to be objectively true or false, depending on whether the associated non-modal proposition is true or false. Thus the question is which features of a ranking function can be objectified in this sense and, in particular, if causation can be so objectified. Spohn defines causation as the transitive closure (pp. 422ff) of direct causation (pp. 354ff) and shows that these two notions can be conditionally objectified (ch. 15). While I remain skeptical that the “natural” or alethic modalities that can be objectified in this sense are rich enough for causation (Huber 2011) and metaphysics in general, I admire the general spirit of the project: to work out in constructive detail and formal precision the positive claims and see how far one gets. Doing so is difficult, and working through it is not easy either. But that is, mainly, because philosophy is difficult.

References Alchourrón, Carlos E. & Gärdenfors, Peter & Makinson, David (1985), On the of Theory Change: Partial Meet Contraction and Revision Functions. The Journal of Symbolic Logic 50, 510-530.

Hild, Matthias & Spohn, Wolfgang (2008), The Measurement of Ranks and the Laws of Iterated Contraction. Artificial Intelligence 172, 1195-1218.

Huber, Franz (2001), Lewis Causation is a Special Case of Spohn Causation. British Journal for the Philosophy of Science 62, 207-210.

Huber, Franz (2007), The Consistency Argument for Ranking Functions. Studia Logica 86, 299-329.

Lewis, David K. (1986), Introduction. In D. Lewis (1986), Philosophical Papers II. Oxford: Oxford University Press, ix-xvii.

Popper, Karl R. (1955), Two Autonomous Axiom Systems for the Calculus of Probabilities. British Journal for the Philosophy of Science 6, 51-57.

Rényi, Alfréd (1955), On a New Axiomatic Method of Probability. Acta Mathematica Academiae Scientiarium Hungaricae 6, 285-335.

Spohn, Wolfgang (1986), The Representation of Popper Measures. Topoi 5, 69-74.

Spohn, Wolfgang (1988), Ordinal Conditional Functions: A Dynamic Theory of Epistemic States. In W.L. Harper & B. Skyrms (eds.), Causation in Decision, Belief Change, and Statistics II. Dordrecht: Kluwer, 105-134.

Spohn, Wolfgang (1993), Causal Laws are Objectifications of Inductive Schemes. In J. Dubucs (ed.), Philosophy of Probability. Dordrecht: Kluwer, 223–52.

Spohn, Wolfgang (2005), Causation: An Alternative. British Journal for the Philosophy of Science 57, 93–119.

Spohn, Wolfgang (2010), Chance and Necessity: From Humean Supervenience to Humean Projection. In E. Eells & J. Fetzer (eds.), The Place of Probability in Science. Boston Studies in the Philosophy of Science 284. Dordrecht: Springer, 101-131.